                         ABSTRACT

The Riemann Hypothesis states that the non-trivial zeros of the Zeta function all have real part one-half. The Hilbert-Polya conjecture states that the imaginary parts of the zeros of the Zeta function corresponds to eigenvalues of an unbounded self adjoint operator.

INTRODUCTION

All attempts failed so far to represent the Riemann duality equation in the critical stripe as convergent (!)  Mellin transforms of an underlying self adjoint integral operator equation.

Two proofs of the RH (P1, P2) are given based on spectral theory in the framework of distributional Hilbert spaces and two appropriately built self adjoint integral operators with related domains. The kernel functions of two convolution integral operators are built as Hilbert transforms of the Theta functions of the Gauss-Weierstrass function f(x) (P1) and the fractional part function (P2).

(P1): The Theta function is defined based on the Gaussian function. The Theta function property is based on the corresponding Poisson summation formula. This property is equivalent to the Riemann duality equation. The non-vanishing constant Fourier term of the standard Theta function jeopardizes the building of a proper self-adjoint operator defining the duality equation.

A Hilbert transformed function does always have a vanishing constant Fourier term. As the Hilbert transform is an isomorphism with respect to the L(2) Hilbert space the original and transformed functions are identical in a weak L(2)-sense.

The Hilbert transform of the even Gaussian function f(x) is given by the odd Dawson function F(x) (which is the derivative of the square of the erf(x)-function) with its appreciated properties (see also "overview").

The Mellin transform M of the Gaussian function f(x) in combination with the singular "Zeta(s)" function (at s=1) defines Riemann's entire Zeta function "Z(s)" by

Z(s) := s*(1-s)*Zeta(s)*M(f)(s) = Z(1-s).

The key idea of (P1) is to replace the M(f)(s) by M(F)(s).

(P2): The key idea of (P2) is to replace the Mellin transform of the fractional part function by the Mellin transform of the Hilbert transform of the fractional part function. This (and not the approach according to (P1) !!) leads to the Riemann duality equation

Z(s)=Z(1-s)

based on an underlying self-adjoint integral operator.

Spectral analysis of convolution integral operators goes along with generalized Fourier analysis for linear, self-adjoint, positive definite operators, defined on appropriate Hilbert space domains. The corresponding domains respectively ranges for P1 and P2 are

P1 Hilbert space:   L(2)-functions with real-axis domain

P2 Hilbert space:   periodic L(2)-functions with (0,1)-domain.

Following this approach the correspondingly built distribution valued holomorphic function ((PeB) p. 38) does have all its zeros on the critical line. In combination with Hardy's theorem ((EdH) 11.1.) that there are infinite zeros on the critical line this proves the RH also in a strong sense.

(P3): Replacing the Gaussian function by the Dawson function also defines an alternative triginometric Zeta function representation, which enables the application of Polya criteria to proof the RH.

(P4+ options): see below

THE SOLUTIONS

A. An alternative trigonometric Zeta function representation built on P1

last update (Note 34): April 1, 2015

B. Two alternative distributional Hilbert space frameworks to answer the RH

last update: Jan 8, 2015

C. An idea of a distributional way to prove the Goldbach conjectures based on P2

A distributional framework is proposed to leverage current Hardy-Littlewood circle method enabling a proof of both, the binary and ternary Goldbach conjectures. The latter one is proven for odd numbers N greater than c*exp(43000), only.

The today´s Fourier analysis of Weyl (periodical, trigonometric) sums in a Banach space framework is proposed to be replaced by a discrete wavelet analysis on the circle in a distributional Hilbert space framework based on the (hypergeometric, non-periodical) Kummer function (WIP).

last update: Jan 8, 2015

THE P1 & P2 SOLUTION CONCEPTS (Oct. 7, 2010; Sept. 11, 2011)

The spectral theory based argument can be applied to both Theta functions, based on the Gauss-Weierstrass function, as well as to the fractional part function. The Hilbert transforms of the Gauss-Weierstrass function and the fractional part function define appropriate kernels of integral operators with corresponding domains and ranges, embedded in appropriate Hilbert spaces: Spectral Analysis to prove the Riemann Hypothesis

The corresponding orthogonal systems of the related Hilbert spaces enable the definition of alternative Fourier integral representations of the Zeta function itself. For the integral operators of P1 and P2 the following polynomial systems are proposed:

P1: The Hilbert transformed Hermite polynomials

P2: The Lommel and/or Bessel polynomials.

THE STEP 1-3 WALKTHROUGH (ISSUE, FRAMEWORK & TOOL, SOLUTION)

The key idea of our two proofs is based on appropriately defined (weak) duality equations of the Zeta function, which are Mellin transforms of corresponding singular, bounded (in appropriately defined Hilbert space framework), self adjoint integral operator equations in a distribution sense. The distributional approach is the prize to be paid to build convergent Mellin transform integrals ((EdH), chapter 10):

STEP 1: ISSUE

An unbounded, self-adjoint integral operator

Domain: The Banach space of continuous functions with max-norm

The constant not vanishing terms of the Theta functions are the root cause of only formally self-adjoint invariant operator definitions with corresponding transform of the Riemann duality equation ((EdH), 10.3). This fact is also reflected in the assumptions of the Müntz formula ((TiE) 2.11). The formula requires additional convergence conditions of the "integral density" function in order to guarantee, that the "inversion ("dual"-1/x variable substitution) operation" is justified. The Poisson summation of the even Gauss-Weierstrass function does not fulfill those conditions. Therefore the Müntz formula cannot be applied to build the Riemann duality equation as transform of an appropriately defined integral operator. The root cause for this is that the Müntz formula is valid in the Banach space framework of the continuous functions equipped with the max-norm.

STEP 2: FRAMEWORK & TOOL

Two bounded, self-adjoint singular integral operators

Domain: Hilbert scale H(-a) with a=0 or a>0

Distributional Hilbert spaces are "elements" of the Hilbert scale H(a) with a<.0. In case of elements u and v of H(0)=L(2) (a=0) with same norm, u and v are identical in a weak H(0) sense, as the L(2) Hilbert space is isomorph to its dual (distribution) space.

The Hilbert transform has the following properties:

- the Hilbert transform of a L(2)-function is again a L(2)-function

- H(b) is densely embedded into H(a) (a<b) with respect to the norm of H(a).

- the Hilbert transform "cuts out" the constant term of a Fourier series representation.

For the relationship between the Theta function property and the Riemann duality equation we refer to:

Mellin & Hilbert transforms

a. Mellin transforms: Let M denote the Mellin transform operator, then it holds

the function f(x) fulfills the Theta property      iff      M(f)(s)=M(f)(1-s).

b. Hilbert transforms: Let H denote the Hilbert transform and g:=H(f), then it holds:

if f fulfills the Theta property, then g fulfills the Theta property in a weak L(2) sense and the constant Fourier term of g is absent.

STEP 3: SOLUTION

Theta property in Hilbert space framework & self-adjoint integral operators

The Hilbert transforms of the Weierstrass function (P1) and the fractional part function (P2) are identical in a weak L(2)-sense with the related originals. Therefore the Theta function property of the original functions keeps valid for the transformed functions in a weak L(2)-sense. This is equivalent to corresponding duality equation representations of the related Mellin transforms (in a weak sense).

As a consequence it holds:

There exist two self-adjoint operators with transform of a complex-valued function Z(s), which is not only formally self-adjoint (in the sense of H. M. Edwards, 10.3: (i.e. there is a "symmetry" with respect to s <--> (1-s) iff there is a variable substitution with respect to x <.-> 1/x)), fulfilling a functional equation in the form Z(s)=Z(1-s) in a weak (distributional) sense. As a consequence all zeros of Z(s) have to lie on the critical line. As the Hilbert transform is idempotent, that is H*H=-I, Z(s) cannot have other zeros than Zeta(s) in a weak sense.

Then, by density arguments in combination with Hardy's theorem the RH is also true in a strong sense.

GENERALIZED FUNCTIONS ON HILBERT SPACES

The corresponding concept is about Hilbert scale theory and Pseudo Differential Operators. For an introduction to this concept we refer to Lifanov, I. K., Nenashev A. S., Generalized functions on Hilbert spaces, singular integral equations, and problems of aerodynamics and electrodynamics

and the references cited there.

DEGENERATED HYPERGEOMETRIC FUNCTIONS

The above solution concept puts the degenerated hypergeometric (Kummer) functions on the stage. The asymptotics of its zeros can be found in

REMARKS AND RELATED TOPICS

Remark 1: The modified Lommel polynomials (Watson G. N., "A treatise on the theory of Bessel functions") are proposed to be the corresponding polynomial orthogonal system as alternative to the today´s Hermite polynomials. D.Dickinson On Lommel and Bessel Polynomials

The positive zeros of the Bessel functions (Polya G., "Ueber einen Satz von Laguerre") are related to a positive representation of polynomials (Polya G., "Ueber positive Darstellung von Polynomen") by the Jensen criterion: Polya G., Ueber die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen

G. Polya based his analysis of zeros of certain entire function on the Theorem of Kakeya, that each polynomial with positive and increasing coefficients have only zeros with absolute value less than 1:

With respect to the Plancherel-Polya theorem we refer to

There is a relationship between a real valued periodic function f(x), its number N(k) of changes of sign of its derivatives in a period and the Hilbert space H(-a), a>0, that f(x) is an entire function:

Remark 2: The vanishing constant Fourier term of the Hilbert transform plays a key role in the above proofs. With respect to the proposed quantum gravity model we note the similarity/affinity to the "cusp form" with its vanishing zero mode in the context of spectral theory on hyperbolic surfaces. Borthwick D., Spectral Theory on Hyperbolic Surfaces

Remark 3: For the Li criterion ("eigenvalue") and Weil ("self-adjoint, positive definite") we refer to the paper from E. Bombieri, J. Lagarias

Let INTEGRAL(f) denote the integral of a real valued function f(x) from x=0 to x=infinite, g(x) be the conjugate of f(x) and x*F(x):=g(1/x). Let further denote M the Mellin transform operator. The multiplicative convolution of f(x) and h(x) is defined by

(f * h)(x):=INTEGRAL(f(x/y)h(y)dy/y.

Then for the Mellin transform of  f * F  it holds:

M(f*F) = M(f)(s) * M(g)(s),

which is real and positive on the critical line Re(s)=1/2.

Theorem: If there is a function f(x) of the type f*F (i.e. if all "involved" integrals are convergent), then this is equivalent to the validity of the Riemann Hypothesis.

The criterion for an appropriate function f(x) given in the paper from E. Bombieri and C. Lagardias ensures those convergent integrals in a strong (continuous function) framework. The concept and theorems of analytical representations of complex-valued Schwartz distributions enables the building of a corresponding function, basically w/o those restricting prerequisites, building on ("just" Hilbert transformed) well known Theta functions.

Remark 4: A change from x to 1/x (especially at x=0,1 and x=infinite) with respect to the measure d(logx)=dx/x with a Theta density function is equivalent to a change of the complex variable s to (1-s) (especially for values s with Re(s)=0,1/2,1) with respect to the corresponding complex valued Mellin transform function.

This also addresses the conceptual issue, when trying to build the continuous analog (in a strong sense) of Euler's ludicrous formula (H. M. Edwards, 10.5).

Remark 5: With respect to the "distributional way to go", we refer to the section/tab "Prime Number Theorem", but also to the physical background of the Berry conjecture:

Remark 6: For the relationships between the Hilbert, Mellin and Fourier transforms with corresponding Mellin multiplier on L(p) in the context of Pseudo Differential Operators, Distributions and complex variables we refer to H. Bremermann, "Distributions, complex variables, and Fourier transforms".

Remark 7: For analytical representation of complex valued Schwartz Distribution (with domain dimension n=1 or n>1) we refer to H. Bremermann (5.9 and 6.6). From B. Petersen, "Introduction to the Fourier Transform & Pseudo-differential Operators" I, §15 we quote: "for a Schwartz distribution Z(s) or Z(1/2+it) ) there exists an analytical function f(z) in the z-plane except on the support K of Z(s), which is in our case the critical stripe resp. line".

We note, that the representation of this analytical function is in line with the pre-requisites for the "concluding remarks" of the paper of E. Bombieri, J. Lagardias.

Remark 8: An interesting generalized Hilbert transform, which is as well of convolution type, in the context of Yukawan Potential Theory and Bessel functions is given in

Duffin R. J., Hilbert transforms in Yukawan Potential Theory, Proc. Nat. Acad. Sci. Vol. 69, No. 12, pp. 3677-3679

This relates to the sections "2nd proof, Jan 2011" and "Quantum Gravity", especially with respect to an alternative modeling of the ground state energy and its related opportunities to a modified quantum mechanics model.

Remark 9: The paper of B. Bagchi is related to our proposed H(-a) Hilbert space solution concept ("2nd proof"). Its Hardy (function) space with its corresponding norm definition might provide the appropriate framework to apply the results and techniques of G. Gasper, "Using sums of squares to prove that certain entire functions have only real zeros".

Remark 10: The proposed mathematical framework (distributional Hilbert spaces and singular integral operators) has been successfully applied in aerodynamics and electrodynamics:

Remark 11:

As an introduction to the Riemann Hypothesis topic we refer to

and just for fun:

Remark 12
: An introduction into the relationship between Fourier analysis/inversion, invariant operators and their transforms, self-adjoint operators with transform Zeta function and the functional equation is given in Edwards H. M.:

Remark 13
: An introduction concerning the analytic character of the Zeta function, its functional equation, as well as to G. Polya´s contribution to "zeros on an integral function represented by Fourier´s integral" is given in

Titchmarsh E. C., "The Theory of the Riemann Zeta-function", Oxford University Press Inc., New York, 1951 (2.1-2.16, 10.23-10.24):

Remark 14: There were some first trials to prove the RH, based on 1st and 2nd kind Bessel functions, which are seen as still promising:

Some related formulas concerning Bessel, Gamma and Zeta functions are summarized in:

The Voronoi summation formula can be regarded as a special case of the Poisson summation formula over RxR. A more symmetrical formula representation applying L(2) theory is given in

The "some formulas concerning ...." paper above is refering to a specific quadratic summation formula containing quadratic 2nd kind Bessel functions with same summand functions (by linear, only) as in the Voronoi formula. The proposition is that this summation representation enables an alternative Voronoi representation in a truly symmetric form, based on a Hilbert space, defined by the Lommel polynomials as orthogonal system.

A modified Voronoi summation formula with quadratic summands would fit to the corresponding representation of the quadrat of the Zeta function Z(s) (i.e. Z(s)*Z(s)) and its related infinite series representation with summands defined by the product of the classical divisor function d(n) and the "Zeta infinite series summands" n*exp(-s).        